A truncated minimal free resolution of the residue field

TL;DR

This paper extends Golod's construction of a minimal free resolution of the residue field for local rings up to degree five, linking it to the algebraic structure of the Koszul complex and providing formulas for measuring Golodness.

Contribution

It generalizes Golod's original minimal free resolution construction to higher degrees and explicitly relates it to the Koszul algebra's multiplicative and Massey product structures.

Findings

Extended minimal free resolution up to degree five.

Explicit formulas for Golodness measurement.

Connection between Koszul algebra structure and resolution construction.

Abstract

In a paper in 1962, Golod proved that the Betti sequence of the residue field of a local ring attains an upper bound given by Serre if and only if the homology algebra of the Koszul complex of the ring has trivial multiplications and trivial Massey operations. This is the origin of the notion of Golod ring. Using the Koszul complex components he also constructed a minimal free resolution of the residue field. In this article, we extend this construction up to degree five for any local ring. We describe how the multiplicative structure and the triple Massey products of the homology of the Koszul algebra are involved in this construction. As a consequence, we provide explicit formulas for the first six terms of a sequence that measures how far the ring is from being Golod.